•
HD28
M414
Dewey
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
Exchange Rate Dynamics with Sticky Prices:
The Deutsch Mark, 1974-1982
by
Alberto Giovannini*
and
Julio
WP# 1522-84
J.
Rotemberg**
February 1984
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
Exchange Rate Dynamics with Sticky Prices:
The Deutsch Mark, 1974-1982
by
Alberto Giovannini*
and
Julio
WP# 1522-84
J.
Rotemberg**
February 1984
*Graduate School of Business, Columbia University, New York
**Massachusetts Institute of Technology, Cambridge, Massachusetts
Revised
November 1983
EXCHANGE RATE DYNAMICS WITH STICKY PRICES:
THE DEUTSCHE MARK, 1974-1982
Abstract
This paper estimates simultaneously dynamic equations for the
Deutsche Mark/Dollar exchange rate and the German wholesale price index,
which emerge from
a
model in which German prices are sticky.
This
stickiness is due to costs of adjusting prices of the form posited by
Rotemberg (1982).
The main results of the empirical analysis are two:
First, the
version of the model where prices are perfectly flexible is rejected.
Second, real exchange rate variability is mostly accounted for by nominal
exchange rate variability.
We find substantial overshooting of the
exchange rate to monetary innovations like those which appear to by
typical in Germany.
Alberto Giovannini
523 Uris Hall
Graduate School of Business
Columbia University
New York, NY 10027
Julio J. Rotemberg
E52-250
Sloan School of Management
Massachusetts Institute of
Technology
Cambridge, MA 02139
irlARlES
APR 1 7 1984
RECEIVED
Revised
November 1983
EXCHANGE RATE DYNAMICS WITH STICKY PRICES:
THE DEUTSCHE MARK, 1974-1982
by
Alberto Giovannini
Columbia University
and
Julio J. Rotemberg
Massachusetts Institute of Technology
1.
Introduction
The idea that purchasing power parity holds with flexible exchange
rates at every moment in time has been shown by the data since 1974 to
be wrong.
Real exchange rates are not constant, and, moreover, their
changes tend to persist (see Frenkel (1981)).
Also, nominal exchange
rates fluctuate much more than price indices.
One possible explanation
for these phenomena, which is advanced in Dornbusch (1976), is that the
prices of goods produced for the domestic market change slowly.
his deterministic model,
a
Then, in
once and for all increase in money leads to an
instantaneous depreciation which "overshoots" the new steady state
exchange rate.
Afterwards, the price level slowly increases and the
This paper is a revised edition of the first chapter of A. Giovannini 's
We
doctoral dissertation at the Massachusetts Institute of Technology.
are grateful to A. Zanello and L. Polman for helping us with various
computational problems, and to L. Dyer for excellent research assistance.
Financial support from the Ente Einaudi and the National Science
Foundation is gratefully acknowledged.
074353G
exchange rate appreciates.
Our paper attempts to estimate
this type on German data since 1974.
slowly.
a
model of
We too assume that prices move
On the other hand, our paper differs in important respects both
from Dornbusch's original formulation and from its empirical
implementation by Frankel (1979) and Driskill and Sheffrin (1981)
.
First, we consider also the impact of nonmonetary variables both on
steady states and on the paths of price levels and exchange rates.
In
particular we focus on the effect of changes in prices of various
imported goods and on changes in real wages.
The latter'
effect on
s
exchange rates has been the focus of work Sachs (1980) among others.
Second, our specification of the stickyness of prices relies on
explicit costs of price adjustment on the part of firms.
These costs of
price adjustment are akin to those in Rotemberg (1982a) and attempt to
capture the fear on the part of firms that customers will desert them if
they follow erratic pricing policies.
Finally, we do not impose the assumption that money is
walk.
a
random
This is simply false for Germany as is noted by Driskill and
Sheffrin (1981).
Hence the response of exchange rates and prices to
monetary shocks differs from the responses considered by Dornbusch (1976)
and his followers.
So, we let the data inform us as to the plausible
Then
stochastic processes followed by the forcing variables we consider.
we assume that exchange rates and prices respond optimally in that
private agents are assumed to also know the stochastic processes of the
forcing variables.
While we feel that these differences constitute important
improvements over previous work we must point out at the outset some
important caveats.
First, the period since 1974 is only partially
a
period of "flexible" exchange rates between Germany and the rest of the
world which we aggregate into
a
dollar region.
This is so because within
Europe exchange rates are only allowed to move within bands.
So, whether
flexible exchange rate models can explain DM exchange rates is an
empirical question.
Second, from the theoretical point of view our model
fails to explicitly take into account the intertemporal budget constraint
that makes the present value of current account deficits equal to the
current stock of net foreign assets.
Unfortunately, explicit use of the
budget constraint appears at the present time to lead to models too
complicated for estimation.
Instead, we assume, as is standard practice
in empirical exchange rate models, that there is perfect asset
substitutability:
that is that the expected yield in domestic currency
of investing at home and abroad is the same.
We do this also for
tractability, since the specification of imperfect substitutability
models is econometrically difficult.
Moreover, the assumption perfect
substitutability, while statistically rejected by several authors, could
be a reasonable approximation of the data.
The structure of the paper is as follows.
model and its so]ution, Section
3
Section
2
presents the
estimates the model using German data,
for the period from June 1974 to February 1983.
Section 4 presents
simulations of the responses of the endogenous variables to
shocks, under the rational expectations assumption.
contains some concluding remarks.
a
variety of
Finally, Section 5
The appendices describe the solution
of the model and the data used in the empirical section.
.
The Model
2.
Our economy is populated by a large number of monopolistically
competitive firms, each producing
good that is differentiated from
a
other domestic goods, and from foreign goods.
Each firm faces the
following demand curve for its product:
where
is demand for good
Q.
i
at time
The first term in the right
t.
hand side of (1) represents substitution between domestic and foreign
goods, and captures substitution both by domestic residents and by
foreigners.
P,
is the index of domestic goods prices,
weighted average of
P.
P
's.
is
the price of foreign consumption goods
each domestic producer does not face
by foreign residents.
a
geometrically
This implies that although our country is "small,"
taken as exogenous.
represents
a
a
perfectly elastic demand schedule
The second term on the right hand side of (1)
real balance effect on domestic demand, Q
foreign activity, and N
is a
stands for
random variable which affects demand for
1
.
goods
On the cost side, we assume that all domestic goods, together with
imported intermediates and labor are used in the production of each good
i.
Marginal costs of production of good
(2)
with W
Q lt
Pp
(1-ot
-a
dt
the nominal wage rate, P M
a
)
w
i
is:
_,_
2
a
t(VW
N
2t
the foreign currency price of imported
When p =
variable affecting productivity.
is a random
intermediate goods, and N„
we have the constant variable costs case.
Real wages relative to the CPI are given by:
W
(3)
where K
is the real
= K
t
CE^) 1
t pJ t
"*
consumption wage at time
is assumed to be
K
t.
This assumption is consistent with
exogenous throughout the paper.
recent empirical findings (von Jurgen (1982)).
Domestic producers are assumed to observe M, P
the time of their pricing decision:
,
P
N
,
Q
E and K at
,
price setting is synchronized.
In the absence of costs of changing prices, domestic producers
would charge
P.
at which marginal cost equals marginal revenue.
,
In
natural logarithms, it is equal to
(4)
Pit
=
lTJ^
+ (1
-
{[pA(Y
K)a
+
" *)
+ a ]e
2
l
1
+
t
VV
-
-
[0(1
+ [p(1
Aa ]p
dt
i
\)(y
-
d) +
(1
"
M(Y
"
d)
Mo^lp^
-
+ a P
+ a k + pdm + Pfq + n
t
t
i t
t*
2 Nt
where lowercase letters are the logs of the variables represented by the
= n„
corresponding uppercase letters, and n
+
Pn^-
However, we follow Rotemberg (1982a) by assuming that monopolists
They are assumed to
also have convex costs of changing nominal prices.
solve the following problem:
00
(5)
Max E _Z
t
Q
P
J
[n(p
it+j
)
-
w.(p.
2
.
t+
-
p.
t+
.)
-
c.(p
it+
.
-
P^^Y
where p stands for
constant discount factor, and
a
is the expectations
E
operator, conditional on information available at time
terms in the square bracket represent
around p.
,
a
second order approximation
of the profit function of producer
of changing prices;
3
i,
in the absence of costs
i
the third term represents the decrease in current
Setting c./w- =
profits to be accounted for by price changes.
every
The first two
t.
c,
for
we have the final expression of the domestic producers'
objective function:
Min E
(6)
t
P
JQ
J
2
[(P it+j
"
P it+j
)
+ c(p.
t+
.
-
2
P it+:hl )
]
Solution of the problem in (6) leads to the equation describing
equilibrium price dynamics for each domestic producer:
/I
<•-,->
(
?
p
t it +
)
"
+ c + pc.
^~
(
l
where for everyJ variable x,
x t±
}
.
t t+j
P it +
_
1
-
1
P it-1 " -p7 P it
p
indicates the expectations of x at
time t+j, conditional on information available at time
Equation (7)
t.
can now be aggregated to obtain the equilibrium dynamics of the domestic
price level:
pA(Y
(8)
tPdt+1
"He"
[1 + C + PC "
1_
pel
-Xl\
p ^dt-1
+
[p(l
-
A)(y
-
d)
+
PY )
(1
d)
"
{[P(1
-
-
-
or
(1
A)(Y
WOjlp^
2
-
A) +
-
1
p^
a9
~Kt
+ PV)
d) +
"
+ a
(1
°
A)a
"
+ a k
1
+ a
l
t
+ pdn>
t
]e
2
t
+ pfq^ +
under the assumption that exogenous variables in (8) are of exponential
order less than 1/p, it can be shown that equation (8) has one stable
n^
and one unstable root and that in this case there exists a unique stable
path that leads to the steady state (Blanchard and Khan (1980)).
domestic price level is
The
function of all future values of exogenous
a
variables, which for domestic producers include m, p
p
,
,
q
,
k,
e
and
n.
The model is closed with the specification of asset markets
equilibrium.
Domestic and foreign bonds are assumed to be perfect
substitutes.
Therefore, the ex-ante real interest differential is zero:
(9)J
e
y
where
t
i
and
i
-
t+1
=
e
t
i
-
i
t
t
are the domestic and foreign interest rate, respectively.
Finally, we have the money demand equation:
m
(10)
where n„
is a random
-
t
Ap
dt
-
(1
-
A)(e
t
+ p
)
ct
= aq
variable affecting velocity.
-
t
bi
t
+ n
3t
In order to obtain
the dynamics of the exchange rate, we need assumptions about equilibrium
output.
The hypothesis is that domestic firms are never rationed in the
labor market, and intermediate goods market, and supply whatever quantity
of the good they produce is demanded.
In this case, equilibrium domestic
output is given by aggregating (1):
=
(11) q
t
(e
t
+ P^ )(Y " d)(l
t
-
A)
-
[v -
(y
-
d)A]p
dt
+ fq^ +dm
+ ti^
t
the dynamics of the exchange rate are obtained by substituting (11) into
money demand, and using the relation (9):
(12)
,
ejl
"
t t+1
ad
where n
= an
-
1
-f^H
+
/b + n„
+
(
-^)(l
i
Pct
+
(1
i^ )(1
a(y
d))]
-
P/ ,
+ a( * " d))
-
H
-
f£ - |( Y -
(y - d)A)
L
dt b
+
r\
+
fi
t
.
This completes the specification of the model.
In matrix form, the
two equations describing the dynamics of the economy can be written as
follows:
(14)
t Y t+1
= Ay
t
+ Bz
t
where:
ct
Nt
't-1
yt
A =
1
P
—
pc
=
m.
t
pA (y
[1
=
Z
-
d)
+ c + r
pc
I [A
-
-
0^(1
1
+ PY
a(Y
A) +
1
-
a,
-
k)a
(Y " d)A)]
-
A)(y
-
[p(l
-
-
d)
+
(1
+
1
pc(l + Py
1
+
1- L^)d
(
b
+ «(Y
"
d)l
cr
2
]
4
p(l
-
\)(y
-
d) +
d) +
-
(1
(1
-
\)a
a.
1,
B =
pc(
(
1^)(1
pc(i + Py)
+ Py)
1
+ a(Y - d)),
o
o
a.
Pd
pc(i + py)'
pc(i + Py)'
ad
af
o
1
pc(i + Py)
-
pc(i + Py)
1
1
b
'
The solution of (14) is:
(16)
p at
dt
=
"'oPdt-i
at i
"
(
L)
r21
j
8
-
J
<r->
1 ( _) z
z w
j
ii
t it+j
it+i
=Q
k=Q 21
[
.
r
]
oo
Iw 1Z
i2
[
22 i=l
1
j=0
(
rJ
22
)J
z
t lt+J
it + i
]
and
LI
s ^I.lCjf-)^^]
fc).
22 i=l
j=0 22
1
(17)
= £<*dt-i-
t
-
where the
uj'
s
,
J
2
z
-) 5:
r 2
(j
J
1Z
(r-)
J
t 1C
Z
i2
it+iJ
22 i=l
j=0 22
and the
4' s
i
are listed in Appendix
1,
J
unstable roots of A and the z.'s are the elements of the
and J
z
are the
vector in
i
(14).
Equations (16) and (17) indicate that the short run dynamics of the
real exchange rate are indeed determined, among other things, by domestic
price setters' expectations about the evolution of variables determining
profits
4
.
For example, given an anticipated future change in the price
of imported intermediate goods, domestic producers would balance the loss
10
of abruptly changing prices when the increase in input costs conies by,
with lower profits today, by partially increasing today's prices.
Short
run fluctuations of the real exchange rate are not exclusively due to
financial disturbances (which would exclusively shift asset demands),
although these shocks play
a
fundamental role here too.
Furthermore,
and more importantly, unanticipated nominal shocks do have significant
real effects, as domestic producers are unwilling to instantaneously
adjust output prices to the long run equilibrium level.
In the long
run, once and for all increases in the money stock can be shown to have
no real effects.
Another important feature of short run equilibrium is that, although
steady state employment is always increased by lowering the consumption
wage, these policies are relatively ineffective in the short run.
Domestic producers do not decrease the relative price of domestic goods
instantaneously, thus giving rise to
with not expansionary effects.
a
short run increase in profits,
In this sense,
short run unemployment can
be considered "Keynesian."
3.
Estimation
This section is divided in two parts.
In subsection a) we present
the estimates obtained from the Euler equations (8) and (12).
b) presents the time series properties of the forcing
variables q
,
i
,
p
,
p„, m and k.
Subsection
11
Exchange Rate and Domestic Price Level Euler Equations
3. a.
The Euler equations we actually estimate are given by:
E (P$
t
3 Pdt + l
(8')
"
Pdt
-
$
+
+
Vdt-1
°
•
k
+
(1
-
(12')
(1
+ p)$
E
t
[e
3
a )p^ +
t
2
?
"
t+1
CI
-
-
<D
1
-
"
$ )e t
7
JL
Ji~
+
pH 3
+
(1
fi
t
1
-
4 )p dt
Vt
fi
<t»
2 p^ t
e
t
1
'Mt
"
JL
+ $ m
'
?
+
«.>
\
+ 5
t
+ $ q^ + n
6
t
t
-
l)p*
]
= °"
]
=
t
where
<t>
V
(
V
(
(1
;
"'
*6 "
and the
(})'s
(
(1
r
l
3
+
<j>
}
+
<t>
2
7
7
+ p)
$3 +
+ P)
«t>
(1 + p)
<t>
3
<t>
"l "
°2
(i + P )
V2
+
3
^
<{.
<t>
+
3
+
7
}
+
2
(J)
<t>
<t>
+
2
7
H.
7
^6
(
(1
+ p)
«J>
+
3
are given in the Appendix.
+
(()
2
}
(|»
2
The price equation is slightly
different from (8) since, for estimation the equation must be
normalized.
The normalization we choose has the advantage of making it
easy to test for the absence of costs of changing prices by testing
whether
<)>
is zero.
Note, however, that without other a priori
restrictions it is not possible to recover all the
the $'s and the a's.
ty'
s
and the a's from
We therefore also reestimate (8') and (12')
12
and a_ are given by .25 and .11 respectively.
assuming that a
These
values are those which are consistent with the estimates of Dramais
(1980).
<h
+
<J>
This, in addition to identifying D which is given by (1 +
(see Table 1), and thus all the
<t>'s,
+
p)(()„
also imposes an additional
constraint.
The residuals obtained when the expectations operators are
eliminated from the left hand side of (8') and (12') have
interpretations.
number of
If technology, money demand and output demand were all
deterministic functions, then n
case, these residuals
variables.
a
and n
would be equal to zero.
In this
would simply be due to innovations in the forcing
That is they would be due to the revisions in expectations
that take place when q^
+1
,
k
t+1>
and P
are realized.
i^+1> P^
t+1
Nt+1
These residuals would thus be uncorrelated with any information available
at time t.
This suggests as
a
natural estimator the instrumental
variables procedure of Hansen (1982) and Hansen and Singleton (1982).
As
Hansen discusses, this procedure is simply three-stage least squares if
one is willing as we are to assume also that the residuals are
conditionally homoskedastic.
This is a particularly reasonable
assumption in our case since our focus on short run fluctuations leads us
to estimate with previously detrended data.
Hansen (1982) also derives
a
portmanteau statistic J which permits the testing of the overidentifying
restrictions.
Unfortunately we cannot seriously entertain the hypothesis
that the cost function and demand functions are deterministic.
and n
Thus n^
also form part of the residuals of (8') and (12').
Insofar as these residuals are independently identically distributed
they are presumably also uncorrelated with certain instruments available
at t.
This would still allow us to use three stage least squares.
13
Another possibility is that the structural residuals of the cost and
demand equations are serially correlated.
in (8') is the sum of two components, e
where u
)
and n
The first, £
.
independently identically distributed.
is
residual in (12') is given by e„
where n
+ n
1r
is the
,
is given by (p n
Then, suppose that n
expectational revision.
u1
Suppose that the residual
+
Similarly the
+ u„
is p„fi
Quasi-
.
differencing (8') and (12'):
pVPdt+z-PiPdt+P
(p
"
dt+r p iPdt )
*
+ [l-(l +P )i3-i 7 -a 2 ](p* t+1 -p lP t )
(8")
VvrPi-V
+
e
(12-)
t+2
-
(4>
1
+
Vvfrf
+
c^-dcp'^^-p^)
+
P2H
£
P^^
+P 2 )e t+1 +
t+l
= £
2t + l
+
f>4 (m
+ U
= £
-
^^-P^)
+ a
+ u
e
it + i
m
"
it+r p i
2
(
PNt + rp/Nt )
it
(l-S-t» A )(P dt+1 -P2 P dt )
t+ rp2 t )
2t + 1
n-d^v^^vfpi^
+
vd^O
+
p 2 £ 2t
Clearly (8") and (12") can still be estimated by three stage least
squares as long as only instruments available at
u
's
are uncorrelated with them.
t
are used and the
However, now the residuals have
a
moving average component.
The estimators are obtained by using two lags for p., e, p
q
,
i
,
k and m as
,
Pvp>
instruments and constraining p to be equal to the
discount factor corresponding to
5
per cent per year.
The monthly data
which we use, from June 1974 to February 1983, is described in
Appendix
2.
The results are reported in Table
1.
Column
I
shows the
14
estimates obtained from (8') and (12').
Column II shows estimates from
the same equation when the a's are constrained.
the results obtained from (8") and (12").
Column III represents
Finally the last column
presents estimates obtained by constraining the a's in (8") and (12").
In columns
I
and II, the coefficient estimates are essentially unchanged
when we constrain the a's.
Quasi-differencing improves the DW statistics
of the exchange rate equation somewhat.
When a's are fixed, however,
the roots do not have the desired property that one is stable while the
The estimates of column II do have this property so
others are unstable.
we use those in our simulations.
The estimates of
<))„
are substantively and significantly different
from zero thus underlining the importance of the price dynamics captured
This significance is undoubtedly due
by the costs of changing prices.
in part to the absence of purchasing power parity in the data.
the other coefficients also have the required signs.
<|>_
and
4> ,
the coefficients of q
,
.
from zero, but have the wrong sign.
ad
(
"
- ).
values of
Most of
The exceptions are
They are insignificantly different
The sign of
<|>,
is the sign of
Other studies (Goldfeld (1973), Rotemberg (1982b)) have found
a
and d below
reported in columns
I
1.
This is consistent with the negative
(J),
and II.
Finally, it must be noted that the J statistic rejects the
overidentifying restrictions imposed by (8') and (12').
This may well
be due in part to our assumption of perfect substitutability between
domestic and foreign assets.
While this assumption may represent
a
good
approximation, it has been statistically rejected by Hansen and Hodrick
(1980) among others.
15
16
Time Series Properties of Exogenous Variable s
3.b.
In this subsection we explore the time series properties of the
forcing variables in order to obtain simple forecasting equations.
These simple forecasting equations can then be attributed to private
This allows us to compute the effects of innovations in the
agents.
forcing variables on the paths of e and p..
First, we consider the vector autoregressions having as variables
p
,
,
e,
q
,
i
,
p
,
p„ and m.
Using monthly data from the middle of 1974
to February 1983, one can accept at the 5% level the hypothesis that q
i
,
p
,
p
,
are not caused in the Granger sense by any other variable
against the hypothesis that they are caused by all the variables in the
system.
The level of money, however, is significantly influenced by the
past values of other variables.
We also considered
money and interest rates are first differenced.
a
system in which
Since this partially
first differenced system is more consistent with univariate evidence
presented below, it probably leads to more reliable tests of hypotheses.
In this second system one can accept for each forcing variable that it is
not caused by any other variable against the hypothesis that they are
caused by all of them.
These results obtain for vector autoregressions
using four, five and six lags.
This doesn't lead us to believe that
these variables are, in fact, independent.
It just means that,
for
forecasting purposes, univariate time series models of these variables
are probably adequate.
This conclusion is tempered somewhat by the fact
that within the vector autoregressions certain individual variables
appear significantly causally prior to other variables.
The two variables which we concentrate on, p, and e, do not cause
the forcing variables in any of the systems we consider.
This is
a
17
Table
2
Significance Levels for the Absence of Causality From X to
Y in Partially First Differenced Vector Autoregressions
VARIABLE Y
VARIABLE X
P XI
p
q
i
m
p„
N
— —
.73
.19
.74
.46
.58
p*
rc
.59
- -
.22
.70
.15
.99
q*
.47
.88
.65
.15
.59
.95
.47
.55
31
m
k
Pd
e
.98
.68
.65
18
precondition for the applicability of the Hansen and Sargent (1980)
formulae.
Table 2 presents the significance levels for the tests that the
coefficients of variable x are zero in the regression explaining variable
y.
These significance levels are obtained from the partially firsts
As can be seen, changes in money are
difference model with six lags.
significantly (at the 5% level) causally prior to changes in the prices
of imported intermediates while changes in interest rates are
significantly causally prior to changes in money.
causal relation on
latter one.
a
We neglect the former
priori grounds while we focus specifically on the
So, we attribute to the agents on section 2 univariate
JL
-'-
forecasting rules for q
k,
,
p
_t_
.A,
,
p„ and
i
.
Instead we assume they
forecast m using also information on past interest rates.
We employ Box-Jenkins techniques to obtain parsimonious
representations for the stochastic processes followed by these variables.
Parsimony is essential here since the forecasting formula of Hansen and
Sargent (1980) becomes very complex as one enriches the parametrization
of these processes.
The application of these principles to monthly data from 1974:7 to
1983:2 lead us to an AR(2) for p" which, when estimated by OLS gives:
4=
1
-
175
4-1
"
(.095)
Similarly p
p"
*ct
(.093)
222
PMt-2
forecasted by an AR(2)
.
p
r
ct-l
.
DW=2.08
R 2 = .93
(0.95)
is also well
=1.247
-
-
.283
(.093)
p
.
„
*ct-2
DW = 2.03
R2 =
.95
:
19
is well explained by just one lag of itself so that:
instead q
q" =
q" .
t_1
.948
t
univariate autoregressions cf
1
suggesting the presence of
for the process followed by
i
-
i
t-1
=
.325
(i
R 2 = .91
(and m) have coefficients which sum to
i
A parsimonious representation
unit root.
a
is:
i
r i t_2
t_
DW = 1.85
-248
"
)
U t _ 2 -i t _ 3
DW = 2.00
)
R 2 = .12
(.094)
(.094)
The most parsimonious process for money which eliminates all significant
serial correlations seems to be:
m
t
"
=
Vi
-
064 (m
t-r m t-2
}
(.093)
-
.086
(
m
t.2
"m
t- 3 )
+
(.097)
+
.068 Ci*.
r i*. 2
)
-
-238 Ci^.
2 -^. 3
)
(.118)
(.113)
+
378 (m -3" m
)
t-4
t
(.096)
-
.023 (i*_ -i*_ )
4
3
(.118)
-
-268 (i*_ -it- 5
4
5
(.113)
DW = 2.00
R2 =
Finally, the real wage k is well forecasted by an AR(3)
k
=
.294 k
(.096)
+
.194 k
(.097)
t_ 2
+
.164 k
(.092)
t_ 3
DW = 2.06
R 2 = .27
.21
.
20
4.
Simulations
In this section, we use the estimates of the parameters of the
mociel
presented in the previous section to simulate responses of the
price level and the exchange rate to
a
variety of shocks.
In the simulations, we substitute the estimates obtained from Euler
equations into the solution of the model, equations (16) and (17).
Furthermore, we apply the Wiener Kolmogorov prediction formulae as in
Hansen and Sargent (1980) to express the infinite summations in (16) and
(17)
in terms only of present and past realizations of the exogenous
variables, using the estimates of the autoregressive parameters of these
variables
These simulations differ from simulations of rational expectations
models as for example in Lipton and Sachs (1980), in that we specify
a
realistic process followed by exogenous variables, rather than implicitly
assuming
a
random walk.
some persistence.
Given our estimates, unanticipated shocks have
Some of the shocks, however, eventually fade away.
By describing the effects of innovations in the processes governing the
exogenous variables, we perform an exercise similar to impulse responses
in vector autoregressive models.
However, these simulations differ from
impulse responses as all the cross equation restrictions arising
from the rational expectations hypothesis are imposed here.
These
restrictions are bound to improve the efficiency of the forecasts under
the rational expectations assumption.
In this sense our simulations can
be thought as constrained impulse responses.
Figures
1
stock of money.
to 4 illustrate the effects of a unit increase in the
Starting from steady state (all endogenous and
21
I
VO
1
00
50
O 00
-0 50
Figure
1
1
:
Money
rJO
O 7S
0.35
-O OS
-0 50
Figure 2: Domestic Prices
(Innovation in Money)
22
>>
o
4 3
2 ?
3
-1 3
Figure 3: Nominal Exchange Rate
(Innovation in Money)
».-»
O
4 3
2 ?
0.3
-1.0
Figure 4: Real Exchange Rate
(Innovation in Money)
23
exogenous variables = 0),
observed.
Figure
1
a
unit disturbance in the money process is
shows that the money stock does increase further
after the shock, and eventually reaches
is due to the fact that,
a
permanently higher level.
as discussed abcve,
This
the data indicates the
presence of nonstationarity of the money stock, and therefore we have
estimated the money process, together with the foreign interest rate
Figure 2 shows the response of the
process, in the first differences.
As stressed above (Section 2), the price level reacts
price level.
immediately to the innovation in the exogenous variable; after the shock,
long run equilibrium is reached in a very smooth fashion, despite the
sizeable swings in the stock of money.
response.
Figure
3
shows the exchange rate
As predicted by the theory, the exchange rate is by far more
volatile than the domestic price level.
The simulations, which use the
estimated parameters, do indeed provide support to the theory outlined
above.
At the time of the increase in the stock of money, the nominal
exchange rate overshoots its steady state level as the theory, following
Dornbusch would predict.
The initial depreciation is four times larger
than the steady state effect on the exchange rate.
The large
fluctuations in the nominal exchange rate generate big swings in the real
exchange rate, as shown in figure
Figures
shock:
5,
5
4.
to 8 illustrate the simulated effects of another financial
an increase in the U.S., treasury bills rate.
As shown in figure
the estimated reaction of German monetary authorities implies that an
increase in U.S. interest rates is followed by
a
large monetary
contraction in Germany.
a
real depreciation of the
This does not prevent
exchange rate as shown in figure
8.
24
10
1
30
0.S0
-0 40
-1 CO
Figure 5: Foreign Interest Rate and Domestic Money
(Innovation in Foreign Interest Rate)
<:-
o
4 3
2.5
0.8
-1 O
Figure 6: Domestic Prices
(Innovation in Foreign Interest Rate)
25
20
21 3
12.5
3 S
-5 e
Figure 7: Nominal Exchange Rate
(Innovation in Foreign Interest Rate)
30
O
21 3
12 5
3.8
-5 ©
Figure 8: Real Exchange Rate
(Innovation in Foreign Interest Rate)
26
Finally, figures
9
to 12 show the effects of a real disturbance,
the form of an increase in the real wage.
in
Here again the exchange rate
responses to the shock is much larger than the price level response, as
the path of the real exchange rate demonstrates (figure 12).
Sone of the
increase in costs is immediately passed into higher prices, which cause
an excess demand for domestic money.
For money market equilibrium, the
domestic interest rate increases relative to the foreign interest rate,
thus implying expectations of exchange rate depreciation, and an
instantaneous appreciation of the exchange rate.
27
J
20
O 24
48
0.11
-0 25
1
II
6
Figure
9:
1*
31
Real Wages
Figure 10: Domestic Prices
(Innovation in Real Wages)
26
31
28
••»
020
-0
015
-0 110
-0 205
-0 300
Figure 11: Nominal Exchange Rate
(Innovation in Real Wages)
100
000
-0.100
-O 200
-O 300
Figure 12: Real Exchange Rate
(Innovation in Real Wages)
29
5.
Concluding Remarks
This paper has specified and estimated a model of sticky prices in
an open economy, under the assumption of rational expectations.
In the
simulations, we have allowed the data to inform us as to the stochastic
processes followed by exogenous variables.
Such
a
model can shed light
on important structural parameters like the relevance of overshooting.
This ability is to some extent independent of whether the model forecasts
well.
Thus we disagree with the view that the poor forecasting
performance illustrated by Meese and Rogoff (1983) completely invalidates
structural estimation of exchange rate models.
It is our opinion that a
large class of exchange rate models, those which do not assume perfect
wage and price flexibility and purchasing power parity, have so far
received little attention from empirical researchers.
For that reason,
it is too early to make general statements about the empirical relevance
of flexible exchange rate theories.
Our empirical analysis is partially successful.
Estimates of the
parameters of the model from the Euler equations do indicate that the
empirical significance of costs in adjusting prices, the fundamental
feature of our model, is considerable.
parameters are very imprecise, and in
However, the estimates of many
a
few cases, of the wrong sign.
This latter phenomenon may be due to some simplifications on which our
model is based, for analytical tractability
:
among these, the exogeneity
of real wages, and the perfect substitutability hypothesis in assets
markets are the most questionable.
Simulations of the model, using the parameter estimates obtained
from Euler equations estimation, and the estimates of autoregressive
.
30
processes for the exogenous variables, show that the data yields
predictions of the effects of disturbances that match the predictions of
the theory.
By far the largest proportion of short run real exchange
rate volatility is accounted for by fluctuations in the nominal exchange
rate:
effects
through this mechanism, nominal disturbances have important real
31
APPENDIX
1
:
The system of equations (14) has a unique convergent solution if
and only if two eigenvalues of A have real parts whose absolute values
are greater than one while the other eigenvalue's real part is smaller
than one in absolute value.
This is so because the system has only one
initial condition namely P Hn
If there are more "stable roots there are
-
infinitely many convergent solutions while on the other hand, the
convergent solution if all roots are explosive may not satisfy the
initial condition.
The solution to the system (14) as long as these
conditions are met uses the Jordan canonical form of A:
-1
12
11
J
12
11
(lxi)
(1X2)
(lxi)
A
A
21
(2xi)
22
(2X2)
r
L
l
(1X2)
r
L
22
(1X2)
2i
(2xi)
C
L
ll
2
is the stable root
C
L
'22
12
21
C
22
(real part less than 1), and J
diagonal matrix containing the unstable roots.
is the
A and the eigenvector
matrix C are decomposed accordingly.
Following Blanchard and Kahn (1980):
where y
'21
= C
C
(16)
12
(2x2)
where
and J
11
(lxi)
'dt
'
r
U
-l
r
L
22 21 P dt-l
"
r
L
-l
T
,-(i+l) r
22.V2
i=0
comes from the partition of B:
L
Z
22^2t t+i
32
B =
~[Y 2 T
We will write equation (16) by expressing all the eigenvectors as
functions of the eigenvalues and the elements of A.
premultiply both sides of (15) by
C,
solve for each of the elements of C.
To do this
normalize the eigenvectors, and
Substituting into (16), the price
and exchange rate equations are:
8
=
p,*
*dt
e
•
t
^ph,
dt-1
"W
t~
i
= to?
_i
dt
dt-i
^
(t-)
J
-
8
oo
)J
z
1
z (
}
(
+
t it
it+j
J
il
J
21
k=Q
21 i=1
-
.
21
=1
li
il
T-
=
UJ
P J 22 J 21
<J.
=
UJ
(J
2
11
4»
+ a (*
2
1
- J
P<D (J
21
22
3
-
21
2
(J
P<t>
gl
21 "
"
(
*1
P4>
(J
3
((
P<t>
'
(
J
~
3
(J
»1
~
p4>
(J
UJ
3
22
2
2
J
)
2
J
2i )
21
22
'
J
)a
22
22
21
)a
J
-
"
22
)a
-
"
22
fl
UJ
31
21
p$ (J 22
3
UJ
"
J
'
(-J
3
)a
i
J
21
}
i
"
J
21
}
J
J
21
)
)
t « (a
1)
22
IX)
ID
-
l)
21
1
)
i
J
z
z
lt+ J
(r-)
t rt+j
J
j=Q 21
where:
J
22
)
(
]
-
rJ
22
}
°°
1
z w
1
J
t it+j
i?
(r-^it-H
i2
=1
j=Q 22
[
.
ii
J
z
i
c-p-) ^
t it+j
(r-)
j
lt+i
iz
j
j=Q 22
22 i=1
]
33
V2
41
"
V*!
"
*J*2
42
P4>
Vg
:
61
at
V*!
I
-
J
(J
22
3
22
-
c+i
J
p<t»
62
p<t)
(J
3
(J
3
J
-
22
21
)
21
)
J
-
22
(<
J
-
"l
p<t)
(
(J
*1
J
-
22
- J
UJ
P 3 (J 22
72
((,
LU
81
(J
p().
3
h7
)
21
)
}
2]
21
)
)
<K
p$ (J 22
3
~"
"
-
*2
(<t>
-
J
-
(^
0,
J
1
21
)
J
21 )(^
J
J
22
21 "
22 )[<t> 2
*11
(J
-
J
21
)[(|)
(J
2
<D
22
(J
2
J
-
-
21
J
<V
(<!>!
•12
)
21
21
}
a
^82
^°
J
-
22
-
J
-
22
3
22
21>
LU
71
}
)
21
j
-
21
21
J
-
p$ 3 (J 22
51
U)
J
3
'
W
-
p$ (J
J
"
22
)
+ a
1)
J
"
22
-
22
1)
-
21
+ a
J
21
)
J
1
21
}
-
(«|»
2
-
((|»
2
1
J
22
)]
)]
34
(<J»
hi
J
-
1
<t>
2
(J
i 22
<t,
2
«31
(»
<t>
(
2
(J
)(<l»
J
-
J
-
22
21
(<t)
^51
J
"
l
J
-
22
21
gl
J
22
J
(»,
*71
-
J
"
J
(
22
"
-
J
V
J
*1
<t)
(
J
21
22 )
)
a
2
a
1
}
V*i
-
V4 V*!
"
2
"
2
<t>
J
"
22
6
($
21
$
J
J
"
22
-
4
7
J
"
22
(0
21
21
21
21 )(^
22
J
(J
2
*1
'
J
22
"
J
J
'
21
)(
22
21
21
»l
• J
22
}
J
-
22
)
}
J
-
21
}
21
22
}
)]
22 )l
)
J
21
21>
21
J
1
'
J
J
)
J
(J
)[
-
"
<t>
22>
J
-
4 72
^81
5
V
l
'62
2
M<i>
'52
*6l"
)
22
5 <D 2
(J
- J
2
22
21
V
Uj
21
)[0>
)(<t)
<t»
=
2
- J
)
21 )(^
22 ai
(J
- J
)
2
22
21
a
1
J
J
"
J
)0,
J
<t»
(<J>
"
1
"
«4l
4 42 =
J
-
22
22
-
21
22
)
21 )(*j
21
(J
1
4 32
J
-
J
J
-
((Jjj
J
1
22
-
((^
)(4>
21
1
}
J
22
)]
}
35
J
'82
x
<j>
2
=
+
= P(l
E
(|)
1
3
3f
A
=
^5
*6
7
J
21
(^-g-^U
\)(y
-
C(l + PY)
_ ad -
-
-
22
r
E Pf
= Pd
1
-
-
a
d) +
(v - d))
(1
\)Oj + a
2
1
,
36
Appendix
2:
The Data
This paper presents an empirical analysis of the dollar mark
exchange rate, but unlike earlier work, aggregates the rest of the world
(for Germany) as a single dollar area.
Data for P
,
is computed using geometrically weighted averages
Q
of the individual series from the countries appearing on table A2
.
1
below.
Table A2
.
Weight
Country
U.S.A.
.157
France
.275
Italy
.184
Netherlands
.252
United Kingdom
.132
The weights are the average from 1974 to 1981 of the ratio of the value
of imports plus exports of each country with Germany over the total
trade of Germany with these countries.
In 1981 the 5 countries in Table
A2.1 represented 45 percent of the total trade of Germany.
countries'
and q are aggregated using the respective
indices for P
exchange rates vis
a
Individual
vis the U.S. dollar.
Most of the data is obtained from the International Financial
Statistics (IFS) tape.
The index of wages in Germany and the index of
intermediate inputs prices are computed using data from the Deutsche
Bundesbank, Monthly Report
,
various statistical supplements.
.
.
37
The exchange rate is the price of
a
U.S.
dollar in Deutsche marks,
line rf in IFS (average over the month)
The domestic price level is line 63 of IFS, the index of wholesale
prices
is computed
P
aggregating the various countries indices of
consumer prices, line 64 in IFS.
P„ is a weighted average of the IFS index of all commodities prices
excluding oil, line 76a, and the index of the dollar price of oil for
Saudi Arabia, line 76aa.
The weights are chosen from the geographical composition of
Germany's imports, using data from the Deutsche Bundesbank Monthly
Report
For the period 1974-1982 we computed the average share of
.
imports from OPEC, in total imports less finished goods.
The average
value share of imports from OPEC is 22 percent.
i
is the U.S.
Q
is the
treasury bill rate, line 60c in IFS.
weighted average of industrial production indices, line
66c, expressed in dollar terms, by dividing each country's index by the
real dollar exchange rate.
M is line 34 in IFS
K is the index of wage costs in industry, per man/hour, divided by
the CPI
.
The former is from Bundesbank Monthly Bulletin
,
the latter is
IFS line 64.
Theory requires that all series be realizations of stationary
stochastic processes.
All series have been demeaned and detrended by
regressing the log of each against
a
constant, time and time squared.
All seasonally unadjusted data has been adjusted by regressing each
series on 11 monthly dummies.
s
38
FOOTNOTES
The presence of real balances in the demand function for good i
Then real money
can be justified by making money yield utility.
balances are a proxy for the marginal utility of income.
In our model,
since money demand is explicitly specified in equation (10), nothing
would be lost if d were zero.
2
3
Constant terms are not reported for simplicity.
This approximation is spelled out in Rotemberg (1982(a)).
4
This feature distinguishes this model from the traditional
Dornbusch (1976) specification. There, the inflation rate depends only
This is due to Dornbusch' s assumption that
on the current value of m.
changes in m last foreever so that current m is the best forecast of
future m.
Driskill and Sheffrin (1981) have interpreted Dornbusch'
continuous time model as saying that p is predetermined at t and thus
doesn't depend on any current variables. For monthly data it is more
plausible, and an equally correct interpretation of the continuous time
model, to assume that p does indeed respond to m
.
Estimation of Euler equations for an exchange rate model with
flexible prices is carried out in Glaessner (1982).
The covariance matrix of the estimates is computed using the
This
technique described in Eichenbaum, Hansen and Singleton (1984).
requires that one obtain the parameters of the vector moving average of
To do
order 1 followed by the products of instruments and residuals.
this we first fit a vector autoregression of order 1 to the product of
instruments and residuals.
Then we explain these products by the lagged
residuals of the vector autoregression. The coefficients of these
This
lagged residuals are treated as the moving average parameters.
procedure is only valid asymptotically as long as the order of the
vector autoregression grows with the number of observations in such a
We use this method
way that, asymptotically, its order is infinite.
because the simpler procedure of Hansen (1982) does not always lead to a
positive definite covariance matrix.
39
REFERENCES
Blanchard, 0. J. and C. M. Kahn, "The Solution of Linear Differential
Models Under Rational Expectations," Econometrica July 1980.
,
"Expectations and Exchange Rate Dynamics," Journal of
Dornbusch, R.
1976.
Political Economy
,
,
Dramais, "Trans-Log KLEM Model for France, Germany, Italy, and the
U. K.," Working Paper No. 41, Mimeo DULBEA, 1980.
Driskill, R. A. and S. M. Scheffrin, "On the Mark:
Economic Review December 1981.
Comment," American
,
Eichenbaum, M.
L. P. Hansen and K. J. Singleton, "A Time Series Analysis
of Representative Agent Models of Consumption and Leisure Choice
Under Uncertainty," manuscript, 1984.
,
A Theory of Floating Exchange Rates
Frankel, J. A., "On the Mark:
Based on Real Interest Differentials," American Economic Review
September 1979.
,
Frenkel, J. A., "Flexible Exchange Rates, Prices, and the Role of 'News':
Lessons from the 1970s," Journal of Political Economy 89, August
1981.
"Formulation and Estimation of a Dynamic Model of
Exchange Rate Determination: An Application of the General Method
of Moments Technique" Mimeo, December 1982.
f7laessner, T. J.
Goldfeld, S. M.
"The Demand for Money Revisited," Brookings Papers on
Economic Activity 3, 1973.
,
,
Hansen, L. P., "Large Sample Properties of Generalized Method of Moments
Estimators," Econometrica July 1982.
,
Hansen, L. P. and R. J. Hodrick, "Forward Exchange Rates as Optimal
Predictors of Future Spot Rates: An Econometric Analysis," Journal
of Political Economy
1980.
,
Hansen, L. P. and K. J. Singleton, "Generalized Instrumental Variables
Estimation of Nonlinear Rational Expectations Models," Econometrica
1982.
Hansen, L. P. and T. J. Sargent, "Formulating and Estimating Dynamic
Linear Rational Expectations Models," Journal of Economic Dynamics
and Control
1980, Vol. 2.
,
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